#508166

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 76 points:

(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,26,41)(16,29,32,63)(17,33,37,62)(18,38,35,23)(21,44,46,30)(22,47,49,36)(25,51,52,74)(27,57,55,73)(28,60,56,43)(31,64,39,48)(34,50,65,45)(40,66,67,53)(42,70,69,59)(54,75,61,71)(58,72,76,68);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,24,41)(16,25,32,52)(17,27,37,55)(18,28,35,56)(21,40,46,67)(22,42,49,69)(23,43,38,60)(29,51,63,74)(30,53,44,66)(31,54,39,61)(33,57,62,73)(34,58,65,76)(36,59,47,70)(45,68,50,72)(48,71,64,75);(2,4)(7,11)(10,12)(13,16,19,32)(14,21,24,46)(15,25,26,52)(17,34,37,65)(18,39,35,31)(20,40,41,67)(22,48,49,64)(23,50,38,45)(27,58,55,76)(28,61,56,54)(29,62,63,33)(30,47,44,36)(42,71,69,75)(43,72,60,68)(51,73,74,57)(53,70,66,59);(5,7,8,11)(6,10,9,12)(15,26)(20,41)(25,52)(27,55)(28,56)(40,67)(42,69)(43,60)(51,74)(53,66)(54,61)(57,73)(58,76)(59,70)(68,72)(71,75)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415362);